Each
of the problems below is intended to be solved by generating a script or
m-file. Be sure to have a very clearly written header for each M-file so that
anyone reading your work can decipher the sort of problems you are solving in
each M-file. Also, be sure to label all axes and assign each plot a title.

A
complete set of m-file solutions can be found here. Of course, your work does not
have to look exactly like these scripts but the results (e.g. plots) should
look much the same. Don’t be lazy and start by looking at these answers
… only refer to these solutions if you get stuck or want to check your
finished work.

In
this problem set we are going to examine MATLAB functions that allow us to
work with polynomials. We'll look at two of the most basic of operations. The
first will focus on plotting out a polynomial function both explicitly and
using POLYVAL. And then calculating the roots of this polynomial. The second
involves fitting a polynomial equation to a given set of data.

For
problems 2-7 submit a single M-file and one plot. Click here to see
what your resulting plot should look like.

Read the HELP
pages on the following functions (either by typing "help" and
then the function name at the MATLAB command prompt, or by clicking
these links): poly,
polyval,
polyfit,
roots.

Generate an
array X
that goes from -4 to 4 in steps of 0.1.

Using X evaluate the
following equation:P
= X3
- 2X2
-4.25X
+ 7.5

and plot P
as a function of X.
Here, use SUBPLOT so that this plot is on the top half of the plot page.

Generate an
array C
that contains the coefficients of this polynomial equation and evaluate
the polynomial using POLYVAL. Remember that polynomial coefficients need
to be arranged in descending order.

Plot the
output of POLYVAL as a function of X. Here, use SUBPLOT so that this
plot is on the bottom half of the plot page.

Use the
coefficient array C
and the ROOTS function to find the roots of the polynomial equation.

On the lower graph place a
symbol (e.g. the letter R
or the symbol X)
at the location of each of these roots. There are many ways to place
this symbol from using the TEXT command to explicitly plotting a single
point. Take your pick and use whatever symbol you'd like but make sure
the root locations are clearly marked.

For the next set of problems we're going to use POLYVAL and POLYFIT in
an attempt to recreate a particular data set. We'll start out with a low
order polynomial and work our way up until we establish a good
reproduction of our original data. Turn in a single M-file and two plot pages (w/
four graphs on each page). Click
here to see what your resulting plots
should look like.

Generate an
array (X)
that goes from -2 to 2 in steps of
0.1.

Plot X as a function
of sin(X)
in one panel of a four panel plot page.

Use POLYFIT to
calculate the coefficients of a polynomial that will describe your data
in X.
Start with a second order polynomial.

Use POLYVAL to
evaluate the polynomial using array X.

On the same
graph that you plotted X
as a function of sin(X)
now plot X
as a function of the array generated when you used POLYVAL. How do they
compare? Can this second order polynomial recreate a close approximation
to your data?

Now let's do
some experimenting. We're going to repeat the last four steps seven(7)
more times. Each time you should plot to a different panel on a 4 panel
plot page. Each time plotting both sin(X) and it's
polynomial equivalent. So by the end of this you will generate two separate plot sheets with
four graphs on each sheet. Be sure to
appropriately label each graph as to the order of the polynomial being
evaluated.